Gödel, Mechanism, Paradox
A famous argument, first proposed in (Lucas 1961), supposedly shows that the human mind has capabilities that go beyond those of any Turing machine. In its basic form, the argument goes like this.
Let S be the set of mathematical sentences that I accept as true. S includes the axioms of Peano Arithmetic. Let S+ be the set of sentences entailed by S. Suppose for reductio that my mind is equivalent to a Turing machine. Then S is computably enumerable, and S+ is a computably axiomatizable extension of Peano Arithmetic. So Gödel's First Incompleteness Theorem applies: there is a true sentence G that is unprovable in S+. By going through Gödel's reasoning, I can see that G is true. So G is in S and thereby in S+. Contradiction!